Many observed time series processes are correlated and also exhibit volatility clustering. Volatility clustering is the tendency of observations relatively small in absolute value to be followed by other small observations, and the tendency of observations relatively large in absolute value to be followed by other large observations. Autoregressive-moving average (ARMA) models, the standard linear time series models for stationary data, cannot alone be used to describe the dependence in these series because ARMA processes with independent and identically distributed errors have constant conditional variances. Nonlinear white noise models with time-dependent conditional variances, most notably generalized autoregressive conditionally heteroskedastic(GARCH) models, are, therefore, often used to describe the ARMA errors of time series with these features. This project addresses ARMA-GARCH model estimation. Specifically, the investigator is researching the distribution for least squares estimators of ARMA model parameters when the white noise error process is dependent and heavy-tailed. A rank-based technique for estimating GARCH model parameters that is both robust and efficient is also being developed. Both standard symmetric GARCH models and asymmetric GARCH models are being considered, and model order selection procedures are being designed.
Correlated processes exhibiting volatility clustering include, for example, atmospheric measurements and economic indicators. In practice, stationary time series of this nature are often first modeled as an ARMA process using least squares and then, if the ARMA residuals exhibit conditional heteroskedasticity, a GARCH model is fit to the residuals. The investigator is developing theory behind this standard procedure for fitting ARMA-GARCH models and developing a robust and efficient GARCH estimation technique. The project results are, therefore, improving the efficiency of model-fitting and enhancing the understanding of corresponding parameter estimates. The overall project impact is a greater understanding of physical and economic systems.
